Centered matrices

CenteredMatrices{𝔽,T} <: AbstractDecoratorManifold{𝔽}

The manifold of $m×n$ real-valued or complex-valued matrices whose columns sum to zero, i.e.

\[\bigl\{ p ∈ 𝔽^{m×n}\ \big|\ [1 … 1] * p = [0 … 0] \bigr\},\]

where $𝔽 ∈ \{ℝ,ℂ\}$.

Constructor

CenteredMatrices(m, n[, field=ℝ]; parameter::Symbol=:type)

Generate the manifold of m-by-n (field-valued) matrices whose columns sum to zero.

parameter: whether a type parameter should be used to store m and n. By default size is stored in type. Value can either be :field or :type.

Y = Weingarten(M::CenteredMatrices, p, X, V)
Weingarten!(M::CenteredMatrices, Y, p, X, V)

Compute the Weingarten map $\mathcal W_p$ at p on the CenteredMatrices M with respect to the tangent vector $X \in T_p\mathcal M$ and the normal vector $V \in N_p\mathcal M$.

Since this a flat space by itself, the result is always the zero tangent vector.

check_point(M::CenteredMatrices, p; kwargs...)

Check whether the matrix is a valid point on the CenteredMatrices M, i.e. is an m-by-n matrix whose columns sum to zero.

The tolerance for the column sums of p can be set using kwargs....

check_vector(M::CenteredMatrices, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the CenteredMatrices M, i.e. that X is a matrix of size (m, n) whose columns sum to zero and its values are from the correct AbstractNumbers. The tolerance for the column sums of p and X can be set using kwargs....

is_flat(::CenteredMatrices)

Return true. CenteredMatrices is a flat manifold.

manifold_dimension(M::CenteredMatrices)

Return the manifold dimension of the CenteredMatrices m-by-n matrix M over the number system 𝔽, i.e.

\[\dim(\mathcal M) = (m*n - n) \dim_ℝ 𝔽,\]

where $\dim_ℝ 𝔽$ is the real_dimension of 𝔽.

project(M::CenteredMatrices, p, X)

Project the matrix X onto the tangent space at p on the CenteredMatrices M, i.e.

\[\operatorname{proj}_p(X) = X - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m x_{j,i}$ for $i = 1, \dots, n$.

project(M::CenteredMatrices, p)

Projects p from the embedding onto the CenteredMatrices M, i.e.

\[\operatorname{proj}_{\mathcal M}(p) = p - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m p_{j,i}$ for $i = 1, \dots, n$.